Merton

Create Merton model object for Vanilla, Asian, Barrier, DoubleBarrier, Lookback, PartialLookback, OneTouch, DoubleTouch, Cliquet, or Binary instrument

Description

Create and price a Vanilla, Asian, Barrier, DoubleBarrier, Lookback, PartialLookback, Touch, DoubleTouch, Cliquet, or Binary instrument object with a Merton model using this workflow:

1. Use fininstrument to create a Vanilla, Barrier, Lookback, PartialLookback, Asian, DoubleBarrier, Binary, Touch, Cliquet, or DoubleTouch instrument object.

2. Use finmodel to specify a Merton model object for the Vanilla, Asian, Barrier, DoubleBarrier, Lookback, PartialLookback, Touch, DoubleTouch, Cliquet, or Binary instrument object.

3. Use finpricer to specify a FiniteDifference, NumericalIntegration, or FFT pricing method for the Vanilla instrument object.

   Use finpricer to specify an AssetMonteCarlo pricing method for the Vanilla, Asian, BarrierDoubleBarrier, Lookback, PartialLookback, Touch, DoubleTouch, Cliquet, or Binary instrument object.

For more information on this workflow, see Get Started with Workflows Using Object-Based Framework for Pricing Financial Instruments.

For more information on the available pricing methods for a Vanilla, Asian, Barrier, DoubleBarrier, Lookback, PartialLookback, Touch, DoubleTouch, or Binary instrument, see Choose Instruments, Models, and Pricers.

Creation

Syntax

MertonModelObj = finmodel(ModelType,'Volatility',volatility_value,'MeanJ',meanj_value,'JumpVol',jumpvol_value,'JumpFreq',jumpfreq_value)

Description

MertonModelObj = finmodel(ModelType,'Volatility',volatility_value,'MeanJ',meanj_value,'JumpVol',jumpvol_value,'JumpFreq',jumpfreq_value) creates a Merton model object by specifying ModelType and the required name-value pair arguments MeanJ, JumpVol, and JumpFreq to set properties using name-value pair arguments. For example, MertonModelObj = finmodel("Merton",'Volatility',0.03,'MeanJ',0.22,'JumpVol',0.007,'JumpFreq',0.009) creates a Merton model object.

Input Arguments

ModelType — Model type

string with value "Merton" | character vector with value 'Merton'

Model type, specified as a string with the value of "Merton" or a character vector with the value of 'Merton'.

Data Types: char | string

Name-Value Arguments

Specify required pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: MertonModelObj = finmodel("Merton",'Volatility',0.03,'MeanJ',0.22,'JumpVol',0.007,'JumpFreq',0.009)

Volatility — Volatility value for the underlying asset

nonnegative numeric

Volatility value for the underlying asset, specified as the comma-separated pair consisting of 'Volatility' and a scalar nonnegative numeric.

Data Types: double

MeanJ — Mean of the random percentage jump size

decimal

Mean of the random percentage jump size (J), specified as the comma-separated pair consisting of 'MeanJ' and a scalar decimal value, where log(1+J) is normally distributed with mean (log(1+MeanJ)-0.5*JumpVol^2) and the standard deviation JumpVol.

Data Types: double

JumpVol — Standard deviation of log(1+J)

decimal

Standard deviation of log(1+J), where J is the random percentage jump size, specified as the comma-separated pair consisting of 'JumpVol' and a scalar decimal value.

Data Types: double

JumpFreq — Annual frequency of Poisson jump process

numeric

Annual frequency of the Poisson jump process, specified as the comma-separated pair consisting of 'JumpFreq' and a scalar numeric value.

Data Types: double

Output Arguments

MertonModelObj — Merton model

Merton object

Merton model, returned as a Merton object.

Properties

Volatility — Volatility value

nonnegative numeric

Volatility value, returned as a scalar nonnegative numeric.

Data Types: double

MeanJ — Mean of the random percentage jump size

decimal

Mean of the random percentage jump size (J), returned as a scalar decimal value.

Data Types: double

JumpVol — Standard deviation of log(1+J)

decimal

Standard deviation of log(1+J), where J is the random percentage jump size, returned as a scalar decimal value.

Data Types: double

JumpFreq — Annual frequency of Poisson jump process

numeric

Annual frequency of the Poisson jump process, returned as a scalar numeric value.

Data Types: double

Examples

Use Merton Model and NumericalIntegration Pricer to Price Vanilla Instrument

This example shows the workflow to price a Vanilla instrument when you use a Merton model and a NumericalIntegration pricing method.

Create Vanilla Instrument Object

Use fininstrument to create a Vanilla instrument object.

VanillaOpt = fininstrument("Vanilla",'ExerciseDate',datetime(2022,9,15),'Strike',105,'OptionType',"put",'ExerciseStyle',"european",'Name',"vanilla_option")

VanillaOpt = 
  Vanilla with properties:

       OptionType: "put"
    ExerciseStyle: "european"
     ExerciseDate: 15-Sep-2022
           Strike: 105
             Name: "vanilla_option"

Create Merton Model Object

Use finmodel to create a Merton model object.

MertonModel = finmodel("Merton",'Volatility',0.45,'MeanJ',0.02,'JumpVol',0.07,'JumpFreq',0.09)

MertonModel = 
  Merton with properties:

    Volatility: 0.4500
         MeanJ: 0.0200
       JumpVol: 0.0700
      JumpFreq: 0.0900

Create ratecurve Object

Create a flat ratecurve object using ratecurve.

Settle = datetime(2018,9,15);
Maturity = datetime(2023,9,15);
Rate = 0.035;
myRC = ratecurve('zero',Settle,Maturity,Rate,'Basis',12)

myRC = 
  ratecurve with properties:

                 Type: "zero"
          Compounding: -1
                Basis: 12
                Dates: 15-Sep-2023
                Rates: 0.0350
               Settle: 15-Sep-2018
         InterpMethod: "linear"
    ShortExtrapMethod: "next"
     LongExtrapMethod: "previous"

Create NumericalIntegration Pricer Object

Use finpricer to create a NumericalIntegration pricer object and use the ratecurve object for the 'DiscountCurve' name-value pair argument.

outPricer = finpricer("numericalintegration",'DiscountCurve',myRC,'Model',MertonModel,'SpotPrice',100,'DividendValue',0.45,'VolRiskPremium',0.09,'LittleTrap',false,'AbsTol',0.5,'RelTol',0.04,'Framework','lewis2001')

outPricer = 
  NumericalIntegration with properties:

                Model: [1×1 finmodel.Merton]
        DiscountCurve: [1×1 ratecurve]
            SpotPrice: 100
         DividendType: "continuous"
        DividendValue: 0.4500
               AbsTol: 0.5000
               RelTol: 0.0400
     IntegrationRange: [1.0000e-09 Inf]
    CharacteristicFcn: @characteristicFcnMerton76
            Framework: "lewis2001"
       VolRiskPremium: 0.0900
           LittleTrap: 0

Price Vanilla Instrument

Use price to compute the price and sensitivities for the Vanilla instrument.

[Price, outPR] = price(outPricer,VanillaOpt,["all"])

Price = 
75.1139

outPR = 
  priceresult with properties:

       Results: [1×6 table]
    PricerData: []

outPR.Results

ans=1×6 table
    Price      Delta        Gamma        Theta       Rho       Vega 
    ______    ________    __________    _______    _______    ______

    75.114    -0.15305    0.00025732    -3.9836    -361.67    4.6317

Use Merton Model and AssetMonteCarlo Pricer to Price Binary Instrument

This example shows the workflow to price a Binary instrument when you use a Merton model and an AssetMonteCarlo pricing method.

Create Binary Instrument Object

Use fininstrument to create an Binary instrument object.

BinaryOpt = fininstrument("Binary",'ExerciseDate',datetime(2022,9,15),'Strike',100,'PayoffValue',130,'OptionType',"put",'Name',"binary_option")

BinaryOpt = 
  Binary with properties:

       OptionType: "put"
     ExerciseDate: 15-Sep-2022
           Strike: 100
      PayoffValue: 130
    ExerciseStyle: "european"
             Name: "binary_option"

Create Merton Model Object

Use finmodel to create a Merton model object.

MertonModel = finmodel("Merton",'Volatility',0.25,'MeanJ',0.02,'JumpVol',0.07,'JumpFreq',0.09)

MertonModel = 
  Merton with properties:

    Volatility: 0.2500
         MeanJ: 0.0200
       JumpVol: 0.0700
      JumpFreq: 0.0900

Create ratecurve Object

Create a flat ratecurve object using ratecurve.

Settle = datetime(2018,9,15);
Maturity = datetime(2023,9,15);
Rate = 0.035;
myRC = ratecurve('zero',Settle,Maturity,Rate,'Basis',12)

myRC = 
  ratecurve with properties:

                 Type: "zero"
          Compounding: -1
                Basis: 12
                Dates: 15-Sep-2023
                Rates: 0.0350
               Settle: 15-Sep-2018
         InterpMethod: "linear"
    ShortExtrapMethod: "next"
     LongExtrapMethod: "previous"

Create AssetMonetCarlo Pricer Object

Use finpricer to create an AssetMonetCarlo pricer object and use the ratecurve object for the 'DiscountCurve' name-value pair argument.

outPricer = finpricer("AssetMonteCarlo",'DiscountCurve',myRC,"Model",MertonModel,'SpotPrice',102,'simulationDates',datetime(2022,9,15))

outPricer = 
  MertonMonteCarlo with properties:

           DiscountCurve: [1×1 ratecurve]
               SpotPrice: 102
         SimulationDates: 15-Sep-2022
               NumTrials: 1000
           RandomNumbers: []
                   Model: [1×1 finmodel.Merton]
            DividendType: "continuous"
           DividendValue: 0
        MonteCarloMethod: "standard"
    BrownianMotionMethod: "standard"

Price Binary Instrument

Use price to compute the price and sensitivities for the Binary instrument.

[Price, outPR] = price(outPricer,BinaryOpt,["all"])

Price = 
53.1178

outPR = 
  priceresult with properties:

       Results: [1×7 table]
    PricerData: [1×1 struct]

outPR.Results 

ans=1×7 table
    Price      Delta     Gamma     Lambda       Rho      Theta     Vega
    ______    _______    _____    ________    _______    ______    ____

    53.118    -0.4432      0      -0.85106    -212.43    1.8604     0  

More About

Merton Jump Diffusion Model

The Merton jump diffusion model [1] extends the Black-Scholes model by using the Poisson process to include jump diffusion parameters in the modeling of sudden asset price movements (both up and down).

The model assumes that the underlying asset price follows a combination of a continuous geometric Brownian motion and jump processes. The continuous component represents the gradual, continuous movement of the asset price, while the jump component accounts for sudden, discontinuous movements caused by unforeseen events.

The stochastic differential equation is

$$\begin{array}{l}d{S}_t=(r-q-{\lambda }_p{\mu }_j)S_{t}dt+\sigma S_{t}d{W}_t+J{S}_{t}d{P}_t\\ \text{prob(}d{P}_t=1)={\lambda }_{p}dt\end{array}$$

where:

r is the continuous risk-free rate.

q is the continuous dividend yield.

W_t is the Wiener process.

J is the random percentage jump size conditional on the jump occurring, where ln(1+J) is normally distributed with mean $$\ln (1+{\mu }_J)-\frac{{\delta }^2}{2}$$ and the standard deviation δ, and (1+J) has a lognormal distribution:

$$\frac{1}{(1+J)\delta \sqrt{2\pi }}\exp \left\{{\frac{-\left[\ln (1+J)-\left(\ln (1+{\mu }_J\right)-\frac{{\delta }^2}{2}\right]}{2{\delta }^2}}^2\right\}$$

where:

μ_J is the mean of J for (μ_J > -1).

δ is the standard deviation of ln(1+J) for (δ≥ 0).

ƛ_p is the annual frequency (intensity) of Poisson process P_tfor (ƛ_p ≥ 0).

σ is the volatility of the asset price for (σ > 0).

References

[1] Merton, R. "Option Pricing When Underlying Stock Returns Are Discontinuous." The Journal of Financial Economics. Vol 3. 1976, pp. 125-144.

Version History

Introduced in R2020a